(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[    158,       7]
NotebookDataLength[ 43447,    1038]
NotebookOptionsPosition[ 42282,    1011]
NotebookOutlinePosition[ 42625,    1026]
CellTagsIndexPosition[ 42582,    1023]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"f", "[", "x_", "]"}], ":=",
RowBox[{"1", "/",
RowBox[{"(",
RowBox[{"1", "+",
   RowBox[{"25",
   SuperscriptBox["x", "2"]}]}], ")"}]}]}], ";"}]], "Input",
CellChangeTimes->{{3.8022915178323207`*^9, 3.8022915288400373`*^9}},
CellLabel->"In[1]:=",ExpressionUUID->"ce68e93b-c806-4724-9b84-e85d10ade32d"],

Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{
RowBox[{"CS", "[",
RowBox[{"n_", ",", "t_"}], "]"}], ":=",
RowBox[{"Module", "[",
RowBox[{
   RowBox[{"{",
   RowBox[{
   "h", ",", "x", ",", "L", ",", "A", ",", "rhs", ",", "i", ",", "z", ",",
      "c", ",", "d", ",", "s"}], "}"}], ",", "\[IndentingNewLine]",
   RowBox[{
   RowBox[{"h", "=",
   RowBox[{"2", "/", "n"}]}], ";", "\[IndentingNewLine]",
   RowBox[{"x", "=",
   RowBox[{"Table", "[",
      RowBox[{
      RowBox[{
      RowBox[{"i", "*", "h"}], "-", "1"}], ",",
      RowBox[{"{",
      RowBox[{"i", ",", "0", ",", "n"}], "}"}]}], "]"}]}], ";",
   "\[IndentingNewLine]",
   RowBox[{"L", "=",
   RowBox[{"Table", "[",
      RowBox[{
      RowBox[{"{",
      RowBox[{
         RowBox[{"x", "[",
         RowBox[{"[", "i", "]"}], "]"}], ",",
         RowBox[{"f", "[",
         RowBox[{"x", "[",
         RowBox[{"[", "i", "]"}], "]"}], "]"}]}], "}"}], ",",
      RowBox[{"{",
      RowBox[{"i", ",", "1", ",",
         RowBox[{"n", "+", "1"}]}], "}"}]}], "]"}]}], ";",
   "\[IndentingNewLine]",
   RowBox[{"TableForm", "[", "L", "]"}], ";", "\[IndentingNewLine]",
   "\[IndentingNewLine]",
   RowBox[{"A", "=",
   RowBox[{"Normal", "[",
      RowBox[{"SparseArray", "[",
      RowBox[{
      RowBox[{"{",
         RowBox[{
         RowBox[{
         RowBox[{"Band", "[",
            RowBox[{"{",
            RowBox[{"1", ",", "1"}], "}"}], "]"}], "\[Rule]",
         RowBox[{"2",
            RowBox[{"h", "/", "3"}]}]}], ",",
         RowBox[{
         RowBox[{"Band", "[",
            RowBox[{"{",
            RowBox[{"1", ",", "2"}], "}"}], "]"}], "\[Rule]",
         RowBox[{"h", "/", "6"}]}], ",",
         RowBox[{
         RowBox[{"Band", "[",
            RowBox[{"{",
            RowBox[{"2", ",", "1"}], "}"}], "]"}], "\[Rule]",
         RowBox[{"h", "/", "6"}]}]}], "}"}], ",",
      RowBox[{"{",
         RowBox[{
         RowBox[{"n", "-", "1"}], ",",
         RowBox[{"n", "-", "1"}]}], "}"}]}], "]"}], "]"}]}], ";",
   "\[IndentingNewLine]",
   RowBox[{"rhs", " ", "=", " ",
   RowBox[{"Table", "[",
      RowBox[{
      RowBox[{
      RowBox[{"(",
         RowBox[{
         RowBox[{"f", "[",
         RowBox[{"x", "[",
            RowBox[{"[",
            RowBox[{"i", "+", "2"}], "]"}], "]"}], "]"}], "+",
         RowBox[{"f", "[",
         RowBox[{"x", "[",
            RowBox[{"[", "i", "]"}], "]"}], "]"}], "-",
         RowBox[{"2",
         RowBox[{"f", "[",
            RowBox[{"x", "[",
            RowBox[{"[",
            RowBox[{"i", "+", "1"}], "]"}], "]"}], "]"}]}]}], ")"}], "/",
         "h"}], ",",
      RowBox[{"{",
      RowBox[{"i", ",", "1", ",",
         RowBox[{"n", "-", "1"}]}], "}"}]}], "]"}]}], ";",
   "\[IndentingNewLine]", "\[IndentingNewLine]",
   RowBox[{"z", "=",
   RowBox[{"LinearSolve", "[",
      RowBox[{"A", ",", "rhs"}], "]"}]}], ";", "\[IndentingNewLine]",
   RowBox[{"z", " ", "=",
   RowBox[{
      RowBox[{"{", "0", "}"}], "~", "Join", "~", "z", "~", "Join", "~",
      RowBox[{"{", "0", "}"}]}]}], ";", "\[IndentingNewLine]",
   "\[IndentingNewLine]",
   RowBox[{"c", "=",
   RowBox[{"Table", "[",
      RowBox[{
      RowBox[{
      RowBox[{
         RowBox[{"L", "[",
         RowBox[{"[",
         RowBox[{
            RowBox[{"i", "+", "1"}], ",", "2"}], "]"}], "]"}], "/", "h"}],
      "-",
      RowBox[{
         RowBox[{"z", "[",
         RowBox[{"[",
         RowBox[{"i", "+", "1"}], "]"}], "]"}], "*",
         RowBox[{"h", "/", "6"}]}]}], ",",
      RowBox[{"{",
      RowBox[{"i", ",", "1", ",", "n"}], "}"}]}], "]"}]}], ";",
   "\[IndentingNewLine]",
   RowBox[{"d", "=",
   RowBox[{"Table", "[",
      RowBox[{
      RowBox[{
      RowBox[{
         RowBox[{"-",
         RowBox[{"L", "[",
         RowBox[{"[",
            RowBox[{"i", ",", "2"}], "]"}], "]"}]}], "/", "h"}], "+",
      RowBox[{
         RowBox[{"z", "[",
         RowBox[{"[", "i", "]"}], "]"}], "*",
         RowBox[{"h", "/", "6"}]}]}], ",",
      RowBox[{"{",
      RowBox[{"i", ",", "1", ",", "n"}], "}"}]}], "]"}]}], ";",
   "\[IndentingNewLine]",
   RowBox[{"s", "=",
   RowBox[{"Table", "[",
      RowBox[{
      RowBox[{
      RowBox[{
         RowBox[{
         SuperscriptBox[
         RowBox[{"(",
            RowBox[{"t", "-",
            RowBox[{"x", "[",
            RowBox[{"[", "i", "]"}], "]"}]}], ")"}], "3"], "/",
         RowBox[{"(",
         RowBox[{"6", "h"}], ")"}]}], "*",
         RowBox[{"z", "[",
         RowBox[{"[",
         RowBox[{"i", "+", "1"}], "]"}], "]"}]}], "-",
      RowBox[{
         RowBox[{
         SuperscriptBox[
         RowBox[{"(",
            RowBox[{"t", "-",
            RowBox[{"x", "[",
            RowBox[{"[",
               RowBox[{"i", "+", "1"}], "]"}], "]"}]}], ")"}], "3"], "/",
         RowBox[{"(",
         RowBox[{"6", "h"}], ")"}]}], "*",
         RowBox[{"z", "[",
         RowBox[{"[", "i", "]"}], "]"}]}], "+",
      RowBox[{
         RowBox[{"c", "[",
         RowBox[{"[", "i", "]"}], "]"}], "*",
         RowBox[{"(",
         RowBox[{"t", "-",
         RowBox[{"x", "[",
            RowBox[{"[", "i", "]"}], "]"}]}], ")"}]}], "+",
      RowBox[{
         RowBox[{"d", "[",
         RowBox[{"[", "i", "]"}], "]"}], "*",
         RowBox[{"(",
         RowBox[{"t", "-",
         RowBox[{"x", "[",
            RowBox[{"[",
            RowBox[{"i", "+", "1"}], "]"}], "]"}]}], ")"}]}]}], ",",
      RowBox[{"{",
      RowBox[{"i", ",", "1", ",", "n"}], "}"}]}], "]"}]}], ";",
   "\[IndentingNewLine]",
   RowBox[{"Return", "[",
   RowBox[{"{",
      RowBox[{"z", ",", "s", ",", "L"}], "}"}], "]"}], ";"}]}],
"\[IndentingNewLine]", "]"}]}], ";"}], "\[IndentingNewLine]"}]], "Input",
CellChangeTimes->{{3.802291530049346*^9, 3.8022915300623384`*^9}, {
3.802291560319058*^9, 3.802291602458931*^9}, {3.802292025462988*^9,
3.80229203316458*^9}, {3.802292104641263*^9, 3.802292179862895*^9}, {
3.8022922142756877`*^9, 3.802292342100909*^9}, {3.802292387653862*^9,
3.8022923949166884`*^9}, {3.802292473444751*^9, 3.8022926573046837`*^9}, {
3.802292887435566*^9, 3.8022930077495184`*^9}, {3.8022930396172915`*^9,
3.802293041318147*^9}, {3.802293092516841*^9, 3.802293093871069*^9}, {
3.802293215367982*^9, 3.802293274009925*^9}, {3.8022933509389067`*^9,
3.8022933531856174`*^9}, {3.802293460977957*^9, 3.802293474149419*^9}, {
3.802293580432829*^9, 3.802293581316324*^9}},
CellLabel->
"In[141]:=",ExpressionUUID->"39e0544f-7073-4850-8a55-89a7f31ee92a"],

Cell[CellGroupData[{

Cell[BoxData[
RowBox[{"CS", "[",
RowBox[{"5", ",", "t"}], "]"}]], "Input",
CellChangeTimes->{{3.8022933424167814`*^9, 3.8022933554433255`*^9}},
CellLabel->
"In[142]:=",ExpressionUUID->"3bcc16c7-6287-4f79-9f2f-50adb96fb456"],

Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"{",
RowBox[{"0", ",",
FractionBox["1020", "247"], ",",
RowBox[{"-",
   FractionBox["945", "247"]}], ",",
RowBox[{"-",
   FractionBox["945", "247"]}], ",",
FractionBox["1020", "247"], ",", "0"}], "}"}], ",",
RowBox[{"{",
RowBox[{
RowBox[{
   RowBox[{
   RowBox[{"-",
   FractionBox["5", "52"]}], " ",

1. Let = + ao and T:=1+ max ail. Oa. Show that if |x| T, then |P(...